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An inner-product Hilbert space formulation of the Kemeny distance is definedover the domain of all permutations with ties upon the extended real line, andresults in an unbiased minimum variance (Gauss-Markov) correlation estimatorupon a homogeneous i.i.d. sample. In this work, we construct and prove thenecessary requirements to extend this linear topology for both Spearman's\(\rho\) and Kendall's \(\tau_{b}\), showing both spaces to be both biased andinefficient upon practical data domains. A probability distribution is definedfor the Kemeny \(\tau_{\kappa}\) estimator, and a Studentisation adjustment forfinite samples is provided as well. This work allows for a general purposelinear model duality to be identified as a unique consistent solution to manybiased and unbiased estimation scenarios.

An unbiased non-parametric correlation estimator in the presence of ties